Near-linear Size Hypergraph Cut Sparsifiers

Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczur and Karger (1996) showed that given any $n$ -vertex undirected weighted graph $G$ and a parameter $\varepsilon\in(0,1)$ , there is a near-linear time algorithm that outputs a weighted subgraph $G^{\prime}$ of $G$ of size $\tilde{O}(n/\varepsilon^{2})$ such that the weight of every cut in $G$ is preserved to within a ( $1\pm\varepsilon$ )-factor in $G^{\prime}$ . The graph $G^{\prime}$ is referred to as a ( $1\pm\varepsilon$ )-approximate cut sparsifier of $G$ . A natural question is if such cut-preserving sparsifiers also exist for hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question and showed that given any weighted hypergraph $H$ where the cardinality of each hyperedge is bounded by $r$ , there is a polynomial-time algorithm to find a ( $1\pm\varepsilon$ )-approximate cut sparsifier of $H$ of size $\tilde{O}(\frac{nr}{\varepsilon^{2}})$ . Since $r$ can be as large as $n$ , in general, this gives a hypergraph cut sparsifier of size $\tilde{O}(n^{2}/\varepsilon^{2})$ , which is a factor $n$ larger than the Benczur-Karger bound for graphs. It has been an open question whether or not Benczur-Karger bound is achievable on hypergraphs. In this work, we resolve this question in the affirmative by giving a new polynomial-time algorithm for creating hypergraph sparsifiers of size $\tilde{O}(n/\varepsilon^{2})$ .